It is the purpose of this tutorial paper to place bispectrum estimation in a digital signal processing framework in order to aid engineers in grasping the utility of the available bispectrum estimation techniques, to discuss application problems that can directly benefit from the use of the bispectrum, and to motivate research in this area Three general reasons are behind the use of bispectrum in signal processing and are addressed in the paper: to extract information due to deviations from normality, to estimate the phase of parametric signals, and to detect and characterize the properties of nonlinear mechanisms that generate time series

Bispectrum estimation: A digital signal processing framework

The strengths and limitations of correlation-based signal processing methods are discussed. The definitions, properties, and computation of higher-order statistics and spectra, with emphasis on the bispectrum and trispectrum are presented. Parametric and nonparametric expressions for polyspectra of linear and nonlinear processes are described. The applications of higher-order spectra in signal processing are discussed. >

Signal processing with higher-order spectra

The bispectrum, which is an ensemble average of a product of three spectral components, is shown to be a very useful diagnostic tool in experimental studies of nonlinear wave interactions in random media. In particular, it is shown that the bicoherence spectrum may be used to discriminate between nonlinearly coupled waves and spontaneously excited waves and to measure the fraction of wave power due to the quadratic wave coupling in a self-excited fluctuation spectrum. Practical aspects of digital bispectral analysis techniques, such as estimation and statistical variability of the estimator, are also discussed. Finally, applications of bispectral analysis techniques in the analysis and interpretation of plasma fluctuation data are described.

Digital Bispectral Analysis and Its Applications to Nonlinear Wave Interactions

Aspects of the nonlinear dynamics of waves shoaling between 9 and 1 m water depths are elucidated via the bispectrum. The biphase values associated with significant bicoherence levels in 9 m depth are consistent with Stokes-like nonlinearities, but as the water depth decreases the waves evolve through a slightly skewed shape somewhat asymmetrical to a vertical axis toward a highly asymmetrical unskewed sawtooth shape. The real and imaginary parts of the bispectrum are the contributions to skewness and asymmetry from individual frequency pairs.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112085003007

Observations of bispectra of shoaling surface gravity waves

The problem of characterizing the sine-wave components in the output of a polynomial nonlinear system with a cyclostationary random time-series input is investigated. The concept of a pure nth-order sine wave is introduced, and it is shown that pure nth-order sine-wave strengths in the output time-series are given by scaled Fourier coefficients of the polyperiodic temporal cumulant of the input time-series. The higher order moments and cumulants of narrowband spectral components of time-series are defined and then idealized to the case of infinitesimal bandwidth. Such spectral moments and cumulants are shown to be characterized by the Fourier transforms of the temporal moments and cumulants of the time-series. It is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts. To put the contributions of the paper in perspective, a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided. >

The cumulant theory of cyclostationary time-series. I. Foundation

Digitally implemented bispectral analysis is used to distinguish between spontaneously excited modes and coupled modes in a self‐excited fluctuation spectrum by measuring the degree of phase coherence between the interacting waves.

E. J. Powers

Papers

Digital bispectral analysis of self-excited fluctuation spectra